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by Michaela Schaumann CERN, Beams Department, Operations Group

Summer Student Lectures 2019. Particle Accelerators and Beam Dynamics Part 3. by Michaela Schaumann CERN, Beams Department, Operations Group. 11.07.2019. A Little recap from yesterday …. Relation between particle momentum , magnetic field and trajectory radius : Beam rigidity.

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by Michaela Schaumann CERN, Beams Department, Operations Group

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  1. Summer Student Lectures 2019 Particle Accelerators and Beam Dynamics Part 3 by Michaela Schaumann CERN, Beams Department, Operations Group 11.07.2019

  2. A Little recap from yesterday … Relation between particle momentum, magnetic field and trajectory radius: Beam rigidity. Particle beam focusing is equivalent to focusing of light with lenses. F = focusing 0 = nothing (dipole, RF, …) D = defocusing 0 Typical alternating sequence of focusing elements. Particle motion through this lattice is described by as an harmonic oscillator. Particles perform betatron oscillations around the design orbit. Number of oscillation in one turn is called tune. The particle’s trajectory through the accelerator can easily be calculated by transfer matrices. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  3. A Little recap from yesterday … Phase space A space that represents all possible states of a system. A particle’s trajectory points or coordinates at a given element draw an ellipse in phase space. The orientation and shape of that ellipse is described by the optical (Courant-Snyder) parameters.  -function The area of that ellipse is emittance. Emittance is a beam property that cannot be changed by focusing. The beam size of a particle ensemble is defined by M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  4. What have we learned so far? We know, how particles behave along the magnetic lattice of an accelerator. • But … • How do we get particles in and out? • How do we accelerate? • How do we get collisions to the experiments? M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  5. Straight Sections and Insertions No bending, no focusing DRIFT space Insertions or straight sections injection kicker extraction kicker transfer line transfer line Arcs • Destination: • next accelerator • target • beam dump injector/ pre-accelerator M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  6. Injection and Extraction • Injection from LINAC3 LEIR – first circular accelerator in for CERN‘s heavy-ions on the way to LHC M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  7. Injection and Extraction Incoming beam Magnetic field No magnetic field Injected beam Circulating beam Septum Magnet Kicker Magnet Extraction follows the same principle, but the beam travels in the opposite direction. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  8. TI8 Injection of Beam 2 into LHC ~ 70 m ~ 3 km ALICE ~ 12 mm court. R. Alemany M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  9. Filling and Circulating Beam • Extraction to PS • Several injections are accumulated, • while the already injected particles circulate/wait. • Only once the ring is fully filled, acceleration starts. • Injection from LINAC3 LEIR – first circular accelerator in for CERN‘s heavy-ions on the way to LHC M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  10. Beam Dump – How to safely kill the LHC beam Schematic of LHC beam dump system LHC beam stores beam ~360MJ energy. Lecture on Accelerator Operation and Design Challenges by A. Lechner Sweep of beam on beam dump window Beam Dump Block (graphite) bunch trains Concrete Shielding ~30cm Beam 8m Source graphics: http://clipart-library.com M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  11. Acceleration acceleration Insertions or straight sections injection kicker extraction kicker transfer line transfer line Arcs • Destination: • next accelerator • target • beam dump injector/ pre-accelerator M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  12. How can we increase the energy of a particle? A charged particles that travels through an electro-magnetic field feels the Lorentz force: Magnetic field B: Force acts perpendicular to path.  Can change direction of particle  cannot accelerate Electric field E: Force acts parallel to path. • Can accelerate • not optimal for deflection = 0 The energy gain of the particle is defined by the integral of the force F over the travelled path dr: M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  13. RF Acceleration and Magnetic Field Increase What about the magnetic field during acceleration? Beam rigidity needs to be increased proportionally to increasing energy.  Machine radius is constant.  Need to increase dipole field accordingly! cavity Circumference 2R + R E+E E 11kA LHC dipole current from 450GeV to 6.5TeV ~20min 750A M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  14. Acceleration without Magnetic Field Increase How does the bending radius changes, when accelerating without adjusting the magnetic field? LHC magnetic dipole field at 450 GeV: Required bending radius at 7 TeV with Binj=0.5T: • Equivalent to 270km circumference • (pure dipole field! without any insertions or quadrupoles) • Magnet surface = 5800km2 • Area of Brunei (South-Eastern Asia) • Area of 2x Luxemburg 43km 2.8km Source:Wikipedia M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  15. Where do we accelerate? Use RF cavities to apply the same accelerating voltage on each passage.  Gradually increase total energy by gaining a small amount each turn. beam B. Salvant N. Biancacci Cavity cavity E-field Example of an accelerating cavity in a synchrotron M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  16. Example: LHC Accelerating System LHC Cavities • LHC has • 8 superconducting cavities per beam • Accelerating field 5MV/m • Can deliver 2MV/cavity • Operating at 400MHz • Beam aperture (radius) ~30cm • Energy gain/turn during ramp 485 keV • (11245 turns/s) R. Calaga, Lecture on Superconducting RF on 17.7.2019 M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  17. RF Acceleration Accelerating voltage is changing with time. That has two consequences: Not all particles see the same voltage, because they arrive at different times. Not all particles gain the same energy. Need synchronization between beam and RF phase to gain energy. There is a synchronous RF phase for which the energy gain fits the increase of the magnetic field. particle bunch Accelerating Voltage Accelerating Voltage Time Time 0 0 negative voltage  deceleration M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  18. Phase Stability (non-relativistic regime) Assume the situation where energy increase is transferred into a velocity increase (non-relativistic regime). Particles P1, P2 have the synchronous phase. field in a cavity unstable stable M2 & N2will go away from P2 unstable (and finally be lost) Longitudinal (phase) focusing keeps particles close to each other ... forming a „bunch“ Ideal particle Particle with Δt < 0 (early) lower energy gain  gets slower Particle with Δt > 0 (late) higher energy gain  gets faster  M1 & N1 will move towards P1stable Courtesy F. Tecker for drawings M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  19. Phase Stability (relativistic regime) Now assume relativistic energies (: An increase in momentum transforms into a longer orbit and thus a longer revolution time. stable unstable cavity M1 & N1will go away from P1  unstable (and finally be lost) E+E Circumference 2R+R E field in a cavity Ideal particle Particle with Δt < 0  higher energy gain  gets longer orbit Particle with Δt > 0  lower energy gain  gets shorter orbit M2 & N2 will move towards P2stable Courtesy F. Tecker for drawings M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  20. Crossing Transition The previously stable synchronous phase becomes unstable when v => c and the gain in path length overtakes the gain in velocity  Transition Transition from one slope to the other during acceleration  Crossing Transition. The RF system needs to make a rapid change of the RF phase, a ‘phase jump’. In the PS: γt is at ~6 GeV, injection at 1.4GeV In the SPS: γt= 22.8, injection at γ=27.7 => no transition crossing! In the LHC: γt is at ~55 GeV, also far below injection energy Transition crossing not needed in leptons machines, why? M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  21. f2 f0 f1 Synchrotron Oscillation Phase space picture Like in the transverse plane the particles are oscillating in longitudinal space. • Simple case (no accel.): B = const. • Stable phase: f0 = 0 • Particle B oscillates around f0. • Particles keep oscillating around the stable synchronous particle varying phase and dp/p. • Typically one synchrotron oscillation takes many turns (much slower than betatron oscillation) • Phase-space ellipse defines longitudinal emittance. • Separatrix is the trajectory separating stable and unstable motion. • Stable region is also called bucket. • Harmonic number h = number of buckets: • fRF = h frev non-rel. regime B stable region separatrix unstable region Courtesy F. Tecker for drawings M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  22. Emittance during Acceleration What happens to the emittance if the reference momentum P0 changes? Can write down transfer matrix for reference momentum change: The emittance shrinks with acceleration! With where g, b are the relativistic parameters. The conserved quantity is It is called normalized emittance. x' px x0’ P0 P1 px0=px1 x1’ x0’>x1’ ps x0=x1 x M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  23. How big are the beams in the LHC? • Normalized emittance at LHC : en= 3.5 mm •  en preserved during acceleration. • The geometricemittance: • Injection energy of 450 GeV: e = 7.3 nm • Top energy of 7 TeV: e = 0.5 nm • The corresponding max. beam sizes in the arc, • at the location with the maximum beta function (bmax = 180 m): • - s450GeV = 1.1 mm • - s7TeV = 300 mm • Aperture requirement: a > 10 s • LHC beam pipe radius: • Vertical plane: 19 mm ~ 17 s @ 450 GeV • Horizontal plane: 23 mm ~ 20 s @ 450 GeV ~ 38 mm ~ 47 mm M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  24. Transverse-Longitudinal Coupling: Dispersion Dipole magnets generate dispersion:  Particles with different momentum are bent differently. Due the momentum spread in the beam , this has to be taken into account for the particle trajectory. Closed orbit for Dispersion function D(s)corresponds to the trajectory of a particle with momentum offset . This also has an effect on the beam size: M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  25. Dispersive Orbit On-momentum orbit in LHC Off-momentum orbit • Dedicated energy (i.e. fRF) change of the stored beam. • Horizontal orbit is moved to a dispersions trajectory. • Vertical orbit unchanged (no vertical dispersion) M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  26. Chromaticity Dipole magnets generate dispersion, which is then focused by quadrupoles. particle having ... to high energy to low energy ideal energy focusing strength Chromaticity Q’ acts like a quadrupole error andleads to a tune spread. Definition of Chromaticity: Q’ measurement at LHC The chromaticity is measured by changing / modulating the energy offset dp/pthrough the RF frequency while recording the tune change DQ. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  27. Accelerator Cycle LEIR Operational Cycle Extraction Injection(s) Energy Beam Intensity Storage/Collisions LHC Operational Cycle Acceleration & increase of magnetic field M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  28. Experiments and Luminosity acceleration Insertions or straight sections injection kicker extraction kicker transfer line transfer line Arcs physics experiment • Destination: • next accelerator • target • beam dump injector/ pre-accelerator M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  29. Experimental Application Each accelerator and experiment requires specific beam properties. Fundamentally different are: Target Secondary particles Beam Fixed Target: Particles that are bent to a circular orbit emit energy/light. Collider: Light Sources: M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  30. ‘Smashing’ Modes and Center-of-Mass Energy The center-of-mass energy defines the upper limit of the newly created particle’s mass. Fixed Target Collider Most of the Energy is lost in the target, only a fraction is transformed into useful secondary particles. All energy is available for the production of new particles. Exercise 2: Derive center-of-mass energy in fixed target and collider experiment. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  31. LHC and its Experiments • In LHC has 4 interaction points (IPs) hosting particle physics experiments: • ATLAS, ALICE, CMS, LHCb • Therefore the two counterrotating beams collide 4 times per turn • When they collide the outer beam cross over to the inner circle and vise versa. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  32. Particle Collisions Experiments are interested in maximum number of interactions per second. The event rate in an experiment is proportional to the luminosity. Interaction Cross-section ( = constant probability for the given interaction to take place) Luminosity Event rate (= number of collisions) Bunch 2 10 11 particles Bunch 1 IP beam size M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019 10 11 particles 32

  33. “Quality Factor” of a Collider The most important measure to describe the potential of a collider is the Luminosity. N….. No. particles per bunch k…… No. bunches f……. revolution freq. g…… rel. gamma …. beta-function at IPs …... norm. trans. emit Defined by the injectors • Overall Goal of an Collider: Maximizing Luminosity! • Many particles (N, k) • In a small transverse cross-section (ε, β) • Performance depends on the injectors: • Production of large N and small ε • Preservation of these parameters until collisions. Limitation: “Collective effects” cause beam instabilities for too high bunch intensities, too small bunch spacing, too “bright” beams. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  34. Optimizing Luminosity Bunch properties (N & ) are defined in the injectors. But what can we do in the Collider? N….. No. particles per bunch k…… No. bunches f……. revolution freq. g…… rel. gamma …. beta-function at IPs …... norm. trans. emit • : defined by the design of the accelerator • F [0,1]: When colliding with many bunches, a crossing angle is needed to avoid unwanted collisions. However this reduces the beam overlap and therefore the luminosity. Keep as small as possible! • Limited by beam-beam effects. • k: Optimize filling scheme and bunch spacing. • : Can be optimized by focusing! Beam 2 Beam 1 M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  35. Mini-Beta Insertions Mini-beta insertion is a symmetric drift space with a waist of the -function in the center of the insertion. Collider experiments are located in mini-beta insertions: smallest beam size possiblefor the colliding beam to increase probability of collisions. There is a price to pay: The smaller b*, the largerb at the triplet. On each side of the symmetry point a quadrupole doublet or triplet is used to generate the waist. They are not part of the regular lattice. l l * β* M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  36. Example: Mini-Beta Insertion at LHC • Example of the LHC • (design report values): • At the interaction point: • b* = 0.55 m • s* = 16 mm • That’s smaller than a hair’s diameter! • At the triplet: • b = 4500 m • s = 1.5 mm = 1500 mm • Largest beams size in the lattice! Matching section + dispersion suppressor Regular arc cells Triplets IP • Limitations: • Tighter tolerances on field errors • Triplet aperture limits b* together with crossing angle. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  37. Squeeze The -functions, and thus beam sizes, in the triplet for small is too large at injection energy  aperture problems.  beam size shrinks with energy  Mini-beta squeezedone at top energy when beam size is smaller. 600m LHC injection optics at 11m x CMS ALICE LHCb ATLAS 10km LHC collision optics at 0.25m in ATLAS & CMS M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  38. Integrated Luminosity What counts for the experiments is not peak performance, but total accumulated number of events: unit “barn” = 10-24 cm2 unit “ inverse barn” = 1024 cm-2 Common order of magnitude: 1 fb-1 = 1039 cm-2 For example: To integrate 1 fb-1 it requires 107 s at For comparison: a year has about × 107 s. LHC has delivered so far ~190 fb-1 to ATLAS/CMS in proton-proton collisions over 7 production years. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  39. “Complete” Collider acceleration Now you know everything to build your own accelerator … … so lets get to work! Insertions or straight sections injection kicker extraction kicker DONE! transfer line transfer line Arcs physics experiment • Destination: • next accelerator • target • beam dump injector/ pre-accelerator M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  40. Everything clear! Hmm …. M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  41. Solutions to Exercises

  42. Exercise 2: Center-of-Mass Energy Exercise 2: Derive center-of-mass energy in fixed target and collider experiment. Center-of-mass (CM) frame is defined where sum of all momenta is zero: 4-momentum can be transformed to center-of-mass frame by Lorentz transformation: Lorentz Transformation The norm: is Lorentz invariant Energy conversation between both frames: M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  43. Exercise 2 : ECM in Fixed Target Experiment M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

  44. Exercise 2: ECM in Collider Experiment Laboratory Frame = CM Frame M. Schaumann, Particle Accelerators and Beam Dynamics, CERN Summer Student Lectures 2019

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